# hsvd

(Not recommended) Hankel singular values of dynamic system

## Syntax

hsv = hsvd(sys)
hsv = hsvd(sys,opts)
[hsv,baldata] = hsvd(___)
hsvd(___)

## Description

hsv = hsvd(sys) computes the Hankel singular values hsv of the dynamic system sys. In state coordinates that equalize the input-to-state and state-to-output energy transfers, the Hankel singular values measure the contribution of each state to the input/output behavior. Hankel singular values are to model order what singular values are to matrix rank. In particular, small Hankel singular values signal states that can be discarded to simplify the model (see balred).

For models with unstable poles, hsvd only computes the Hankel singular values of the stable part and entries of hsv corresponding to unstable modes are set to Inf.

hsv = hsvd(sys,opts) computes the Hankel singular values using options that you specify using hsvdOptions. Options include offset and tolerance options for computing the stable-unstable decompositions. The options also allow you to limit the HSV computation to energy contributions within particular time and frequency intervals. See balredOptions for details.

[hsv,baldata] = hsvd(___) returns additional data to speed up model order reduction. You can use this syntax with any of the previous combinations of input arguments.

hsvd(___) displays a Hankel singular values plot.

## Examples

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Create a system with a stable pole very near to 0, and display the Hankel singular values.

sys = zpk([1 2],[-1 -2 -3 -10 -1e-7],1);
hsv = hsvd(sys)
hsv = 5×1
105 ×

1.6667
0.0000
0.0000
0.0000
0.0000

Notice the dominant Hankel singular value with magnitude $1{0}^{5}$, which is so much larger that the significant digits of the other modes are not displayed. This value is due to the near-unstable mode at $s=1{0}^{-7}$. Use the 'Offset' option to treat this mode as unstable.

opts = hsvdOptions('Offset',1e-7);
hsvu = hsvd(sys,opts)
hsvu = 5×1

Inf
0.0688
0.0138
0.0024
0.0001

The Hankel singular value of modes that are unstable, or treated as unstable, is returned as Inf. Create a Hankel singular-value plot while treating this mode as unstable.

hsvd(sys,opts)
ans = 5×1

Inf
0.0688
0.0138
0.0024
0.0001

The unstable mode is shown in red on the plot.

By default, hsvd uses a linear scale. To switch the plot to a log scale, right-click on the plot and select Y Scale > Log. For information about programmatically changing properties of HSV plots, see hsvplot.

Compute the Hankel singular values of a model with low-frequency and high-frequency dynamics. Focus the calculation on the high-frequency modes.

Load the model and examine its frequency response.

bodeplot(Gms)

Gms has two sets of resonances, one at relatively low frequency and the other at relatively high frequency. Compute the Hankel singular values of the high-frequency modes, excluding the energy contributions to the low-frequency dynamics. To do so, use hsvdOptions to specify a frequency interval above 30 rad/s.

opts = hsvdOptions('FreqInterval',[30 Inf]);
hsvd(Gms,opts)
ans = 18×1
10-4 ×

0.6237
0.4558
0.3183
0.2468
0.0895
0.0847
0.0243
0.0028
0.0000
0.0000
⋮

## Tips

To create a Hankel singular-value plot with more flexibility to programmatically customize the plot, use hsvplot.

## Compatibility Considerations

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Not recommended starting in R2021a